A Mathematical Model for Predicting the Extrusion ...

ISSN: 2276-7835

ICV: 5.62 Submitted: 08/11/2015 Accepted: 31/12/2015 Published: 03/02/2016

DOI: http://doi.org/10.15580/GJSETR.2016.1.110815153

A Mathematical Model for Predicting the Extrusion Efficiency of a Vertical Column Shaft Palm Oil Extrusion Machine using Dimensional Analysis By

Okafor, Victor Chijioke, Ejesu, Princewill Kelechi, Chinweuba, Dennis Chukwunonye Nwakuba, Reginald Nnaemeka

Greener Journal of Science Engineering and Technological Research

ISSN: 2276-7835 Vol. 6 (1), pp. 001-008, February 2016.

Research Article (DOI http://doi.org/10.15580/GJSETR.2016.1.110815153)

A Mathematical Model for Predicting the Extrusion Efficiency of a Vertical Column Shaft Palm Oil Extrusion Machine using Dimensional Analysis Okafor, Victor Chijioke1, Ejesu, Princewill Kelechi2, Chinweuba, Dennis Chukwunonye3* and Nwakuba, Reginald Nnaemeka4 1

Department of Agricultural and Bioresource Engineering, School of Engineering and Engineering Technology, Federal University of Technology, Owerri Imo state Nigeria. Email: [email protected] 2 Department of Agricultural and Bioresource Engineering, School of Engineering and Engineering Technology, Federal University of Technology, Owerri Imo state Nigeria. 3 Department of Agricultural and Bioresource Engineering, School of Engineering and Engineering Technology, Federal University of Technology, Owerri Imo state Nigeria. 4 Department of Agricultural and Bioresource Engineering, School of Engineering and Engineering Technology, Federal University of Technology, Owerri Imo state Nigeria. Corresponding Author’s E-mail: dennischukwunonye@ gmail. com ABSTRACT This research is based on the development of a mathematical model for predicting the extrusion efficiency of a vertical column shaft palm oil extrusion machine using dimensional analysis based on the Buckingham theorem. An equation/algorithm for the extrusion efficiency of a vertical column shaft palm oil extrusion machine using the Buckingham theorem was generated and also the efficiencies obtained by the predicted equation was compared with that of the experimented efficiencies at different revolution speeds per minute (rpm). A high coefficient of determination of about 82.4% between the predicted and experimental values showed that the method is good. The model was validated with data obtained from an operational extrusion machine and there was no significant difference between the experimental extrusion efficiency and the predicted extrusion efficiency respectively at 5% degree of freedom. Keywords: column, shaft, extrusion, oil, Buckingham, model.

INTRODUCTION Oil palm which is an ornamental and economically valuable palm tree has its nativity to western Africa and is widespread throughout the tropics. The oil palm grows up to 9m (30ft) in height. It has a crown of feathery leaves that are up to 5m (15ft) long. The flower cluster is on a short thick spike at the base of the leaves. Flowering is followed by the development of a cluster of egg-shaped, red, orange, or yellowish fruits. Each fruit is approximately 3cm (1in) long and contains from one to three seeds embedded in a reddish pulp (shefh et al., 2002). Palm oil is extracted from the fruit pulp. This yellowish or reddish oil is mostly used in the manufacture of soap and candles. Palm oils also the largest source of palmitic acid, a fatty acid used in numerous commercial processes (Mark, 1998). The following steps are conventionally applied in the extraction of palm oil from its fruit pulp; at first the palm fruit is being dehausted from its bunch after which it is parboiled to loosen the tissues. The purpose of parboiling is to reduce toxicity, acidity, and alkalinity and also bring about oxidation of the palmitic acid. Engineering is of the view that parboiling helps in increasing the extrusion efficiency by softening the palm fruit tissues. After the parboiling process, the palm fruit is then introduced into the vertical column extrusion machine where the oil is extracted and separated from the tissue chaff. This research is therefore based on the development and prediction of a mathematical model for determining the extrusion efficiency of a vertical column shaft palm oil machine using the Buckingham theorem.

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Most of the research work is tailored into modeling of parameters which determine the functionality of an extrusion machine. Most of these models are specific and are related to a particular design. Some researches (Degrimencioghu, Srivastava 1996, Shefh et al., 1996; Mohammed 2002; Ndrika 2006) used the dimensional analysis based on the Buckingham theorem as veritable instrument in establishing a prediction equation of the various systems. According to Douglas et al., 2003, there are basically two methods employed in mathematical modeling – indicial and group ( ) methods. The indicial method is rather lengthy if there are a larger number of variables. It is therefore necessary to develop a more generalized methodology that would lead directly to a set of dimensionless grouping whose numbers would be determined in advance by a scrutiny of the matrix formed from the variables considered to be relevant to the investigation and the relevant dimensions necessary to describe those variables. Such a technique is known as the Buckingham theorem.

MATERIALS AND METHOD Description of the Prototype Model of the Vertical Column Shaft Palm Oil Extrusion Machine A vertical column shaft palm oil extrusion machine described by Ohanele (2005) is used in validating the model. The machine is powered by a 1,200KW electric motor and operates with a centrifugal action at a relatively moderate revolution speed of about 3 – 6 rpm. The machine consists of a conical shaped hopper that is open into a chamber through which the parboiled palm fruit is introduced into the extrusion chamber. A vertical column shaft of about 18 – 22cm in diameter with a V- shaped threads is fitted into the extrusion chamber; this shaft is connected to the 1200KW electric motor through a bevel gear which has the capacity to redirect a vertical motion to a horizontal motion and vice versa. The centrifugal action of the shaft meshes with the palm fruit hence compressing them which in turn breaks the tissues, afterwards the oil is extracted. The speed of the shaft is relatively low so as to avoid cracking of the kernel nut. The oil is collected under the influence of gravity through a conical shaped chute. Model Development Extrusion involves all action from the hopper orifice through the extrusion chamber to the collector chute (Asoegwu et al., 2010). The physical quantities affecting the extrusion processes include both the crop physical properties and machine parameters (Ndukwu 1998; Simonyan et al., 2006; Asoegwu et al., 2010). The crop properties include: crop species, age, moisture content, bulk density, palm fruit geometric mean diameter, and that of the machine properties are the feed rate, diameter of the extrusion chamber, shaft speed, throughput capacity and the pitch of the thread fitted to the vertical column shaft. In course of the model development, the following assumptions were made (Asoegwu, 2011): 1. 2. 3. 4. 5. 6. 7.

The moisture content of the palm fruit is approximately the same, The palm fruit dimension is constant at the same moisture content, The thickness of the fruit is the same at the same moisture content, Diameter of the extrusion chamber is fixed, Distance between the channel and extrusion chamber is fixed, The age of the palm fruit is the same, and, The individual weight and the volume of the fruit is constant at a particular moisture content,

Based on the assumptions, the major variables of importance are: the palm fruit moisture content, bulk density of the palm fruit, palm fruit particular density, feed rate, throughput capacity, and extrusion speed (Ndukwu, 1998). The extrusion efficiency which is a function of meshed and unmeshed palm fruit recovered from the collector chute is given as:

η

ex

=

(

f φ ,σ 1, σ 2 , γ ,ν , ω ,τ c r

)

1

Where:

η

ex

is the extrusion efficiency (%),

φ is the palm fruit moisture content (%), www.gjournals.org

2


σ σ γ


3

Is the bulk density of the palm fruit (Kg/m ),

1

3

2

Is the particle density of the palm fruit (Kg/m ),

Is the feed rate (Kg/s)

r

ν Is the revolution speed per minutes (rpm) ω Is the diameter of the extrusion chamber (m)

τ

c

Is the machine throughput capacity (Kg/s)

The number of dimensionless groups arising from a particular matrix formed from ‘n’ variables in ‘m’ dimensions is given as (n-r) where ‘r’ is the largest non-zero determinant that can be formed from the matrix and therefore the equation relating the variables will be of the form

f (π 1 ,π 2 ,π 3 , ..........π n− r ) = 0

2

From equation one, the number of variables of importance that determine the extrusion efficiency of the palm fruit extrusion machine ‘m’ is seven (7) and the number of fundamental units ‘n’ is three (3), therefore the number of

π terms ( π n ) is given as:

π

n

=m–n=7–3=4

Hence we have the

π = π ,π , π 1

n

2

3

π

group as:

and π 4

The dimensional matrix of the parameters is shown in table 1. Table 1: the dimensional matrix of the variables Dimensions M L T

φ 0 0 0

σ

σ

1

1 -3 0

γ

2

1 -3 0

τ

r

1 0 -1

υ

ω

0 1 -1

0 1 0

c

1 0 -1

According to Douglas et al., 2003, independent dimensionless groups are defined as those which can be formed from any particular number of quantities, but are independent of each other in the sense that none of them can be formed by any combination of the others. The number of repeating variables is therefore given as (m variables and the number of dimensionless

π groups

variables is (7 – 4) being three (3). By repeating dimensionless groups featuring

τ π =γ υ π = σγ ω υσ ω = π γ

φ , γ , υ , and ω r

n

) where ‘m’ and ‘

π

n

' are the total number of

respectively. Hence, for this study the number of repeating

τ ,σ c

π

1

, and σ 2 it follows that it will be necessary to seek the

. Hence we have:

c

3

1

r

2

1

4

2

r

2

2

5

3

r

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3


π

4


=φ

6

By modeling, we have it that the predicted extrusion efficiency is given as:

ηp

ex

= f (π 1 ,π 2 ,π 3 ,π 4 )

Substituting the values of

7

π , π ,π 1

2

3

, and π 4 into equation 7, we have

2 2   p =  τ c , υ σ 1ω , υ σ 2 ω , φ  η ex  γ γ γ r   r r

8

According to shefh et al., 1996, we can combine the dimension terms to reduce them to a manageable level either by multiplication or division or by both.

π π

1

τ

=

c

9

υ σ 1ω 2

2

ππ 3

4

=

υ σ 2 φω 2

γ

10

r

Hence equation 8 is reduced to

η p = (π ;π ) ex

12

11

34

Substituting equations 9 and 10 into equation 11 we have 2   p =  τ c ;υσ 2φ ω  η ex  υ 2 γ r  σ ω 1 

12

Development of the Prediction Equation/Algorithm The prediction equation is established by allowing one of the π terms to vary at a time while keeping the other constant and observing the resulting changes in the functions (shefh et al., 1996). This is achieved by plotting the values of the extrusion efficiency against the π terms. Based on the equation of a straight line graph (y=Mx + K), we have it that

ηe

ex

= 17.22π 12 + 44.9

13

Where: 44.9 is the intercept on the y axis, 17.22 is the slope of the graph, and,

π

12

is the dimensionless parameter of the x axis as shown in figure 2

Based on the equation of a straight line graph (y = Mx + K) we have it that

ηe

ex

= − 18.71π 34 + 76.98

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Where: 76.98 is the intercept on the y axis, -18.71 is the slope of the graph, and,

π

is the dimensionless parameter on the x axis.

34

The plot of the π terms (i.e. Figures 1 and 2) according to Mohammed (2002) forms a plane surface in linear space. It means that their combination favours summation or subtraction. Shefh et al., 1996 is of the view that the component equation is formed by the combination of equations 13 and 14 respectively.

ηp

= ex

f (π ,π )− f (π ,π )+ K 1

12

34

2

12

15

34

Where:

f ,π

At

1

34

is kept constant while

By substituting the parameters of

η

ex

π is varied, and, at f ,π is kept constant while π is varied. f (π ,π )and f (π ,π ), equation 15 is written as: 12

1

12

2

12

34

2

12

34

34

=17.22π 12 + 44.9 − (− 18.7 π 34 + 79.98)

16

Therefore the prediction equation becomes

ηp

ex

=17.22π 12 + 18.71π 34 − 79.98

By substituting the values of the dimensionless

17

terms (i.e.

π

12

and π 34) , we have the predicted extrusion

efficiency as: 2 υ    φ τ σ  c 2 ω    + 18 . 71 − 35.08 η ex =17.22 υ 2   γ  σ 1ω   r 

p

18

Figure 1: A graph of experimental extrusion efficiency dimensionless

π

12

keeping

π

34

η

e ex

against

constant

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Figure 2: A graph of experimental extrusion efficiency

π

34

with

π

12

ηe

against ex

kept constant

Determination of the Validation Parameters According to Khurmi and Gupta (2004), the experimented extrusion efficiency of a vertical column shaft palm oil extrusion machine is given as:

ηe

= ex

Ideal Effort x 100 Actual Effort

19

Where the ideal effort (Po) and actual effort (P) are given as:

P

o

ηe

= W tan α , and, P = W tan(α + ϕ ) hence the experimented extrusion efficiency is given as: = ex

Where

tan α x 100 tan(α + ϕ )

20

α is the helical angle of the threaded vertical shaft ( o ),

ϕ is the friction angle between the palm fruit and the shaft thread ( o ) ψ But tan α = Πd

21

Where :

ψ is the pitch of the screw thread , and , d is the mean diameter of the screw. tan α + tan ϕ But tan (α + ϕ ) = 1 − tan α tan ϕ tan ϕ is the coefficient of friction between the screw and the palm fruit.

22

RESULTS AND DISCUSSION The mathematical model was validated using data generated from an existing palm oil extrusion machine owned by SIAT Nigeria Limited (Former Rison palm Nigeria Limited) for a period of five years. The model validation was done at four different revolution speeds per minutes (rpm) and constant feed rate as shown in table 2. Microsoft Excel 2007 statistical package for window vista/window 7 was used for the graphical analysis and the predicted and experimental extrusion efficiencies are shown in table 3. www.gjournals.org

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Based on the equation of a straight line graph we have it that

η

p ex

= 0.573η e + 60.12

23

ex

Where:

η η

p ex e ex

is the predicted extrusion efficiency (%) , and, is the exp erimented extrusion efficiency (%)

From figure 3, it can be observed that the predicted and experimented extrusion efficiencies have a very high correlation value of about 82.4% Table 2: Evaluation parameters for the Extrusion Machine at four Revolution Speeds per Minute Parameters First Speed Second Speed Third Speed Fourth Speed Feed rate (Kg/s) 700 700 700 700 Revolution Speed 3.92 4.43 4.98 5.21 per Minutes (rpm) Bulk density 823.1 834.2 834.64 844.57 (Kg/m3) Throughput 652 642 646 651 capacity (Kg/s) Diameter of 150 150 150 150 extrusion chamber/ring (m) Particle density 433 437 501 509 Moisture content 10.94 11.32 11.74 13.84 Table 3: Experimental and Predicted Extrusion Efficiencies for four different Revolution Speeds per Minutes Revolution Speed per Minutes Experimental Efficiency Predicted Efficiency (RPM) 3.92 58.21 60.42 4.43 60.52 62.84 4.98 61.36 63.41 5.21 61.94 65.03

Figure 3: A graph of experimental extrusion efficiency against predicted extrusion efficiency

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CONCLUSION A mathematical model for the determination of the extrusion efficiency of a vertical column shaft palm oil extrusion machine was predicted using dimensional analysis based on the Buckingham’s theorem. A functional relationship between the predicted and experimented equations shows a high level of coefficient of determination which implies good agreement. Also there was no significant difference between the experimented and predicted extrusion efficiencies at 5% degree of freedom. REFERENCES Asoegwu S.N, Agbetoye L.A.S., and Ogunlowo A.S., (2010). Modeling flow rate of Egusi – melon (Colocynthis citrullus) through circular horizontal hopper orifice. Advance in Science and Technology, 4:35 – 44. Degrimencioglu A. and Srivastava A.K., (1996). Development of screw conveyor performance models using dimensional analysis. Transactions of the ASABE, 39:1757 – 1763. Khurmi R.S. and Gupta J.K. (2004). Machine Design. Eurasia, New Delhi, India. Mark O.L. (2012). Technical review on the morphological properties of palm fruit. International journal on agricultural science, 23: 112 – 129. Mohammed U, S. (2002). Performance modeling of the cutting process in sorghum harvesting. (PhD Thesis.), Zaria, Ahmadu Bello University. Ohanele W.C., (2006). Performance evaluation of a vertical column palm oil extraction machine. (B. Eng. Thesis.), Port Harcourt, University of Port Harcourt. Ndrika V.I.O. (2006). A mathematical model for predicting output capacity of selected stationary grain threshers. Agricultural mechanization in Asia, Africa and Latin America, 36: 9 – 13. Ndukwu M.C and Asoegwu S.N, (2011). A mathematical model for predicting the cracking efficiency of vertical shaft centrifugal palm nut cracker. Research in Agricultural Engineering, 57: 110 – 115. Ndukwu M.C. and Asoegwu S.N., (2010). Functional Performance of a vertical shaft centrifugal palm nut cracker. Research in Agricultural engineering, 56: 77 – 83. Shefh S., Upadhyaya S.K., and Garret R.E., (1996). The importance of experimental design to the development of empirical prediction equations: A case study. Transaction of ASABE, 39: 377 – 384. Simonyan K.J., Yilijep Y.D., and Mudiare O. J., (2010). Development of a mathematical model for predicting the cleaning efficiency of stationary grain threshers using dimensional analysis. Applied engineering for agriculture, 26: 189 – 195. Cite this Article: Okafor V.C., Ejesu P.K., Chinweuba D.C. and Nwakuba R.N. (2016). A Mathematical Model for Predicting the Extrusion Efficiency of a Vertical Column Shaft Palm Oil Extrusion Machine using Dimensional Analysis. Greener Journal of Science Engineering and Technological Research, 6 (1): 001-008, http://doi.org/10.15580/GJSETR.2016.1.110815153.

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